Expectation of multivariate log normal

Question

I am trying to compute the following:

Assume that $(a, b, c)$ are jointly normal with unknown means $(\mu_a, \mu_b, \mu_c)$ and variance-covariance matrix $\Sigma$.

What is $E[exp(a)|a>0, b>0, c>0]$?

When I input this into Mathematica, it hangs and fails to produce a result. My sense is that this is because I'm not telling it that $\Sigma$ is positive-definite (obviously it can tell that $\Sigma$ is symmetric).

My code is

Expectation[Exp[a] \[Conditioned] {a > 0, b > 0, c > 0},
 {a, b, c} \[Distributed] 
  MultinormalDistribution[{μa, μb, μc},
   {{σaa, σab, σac}, {σab, σbb, σbc}, {σac, σbc, σcc}}]]

Following the suggestion of using TruncatedDistribution, I tried running the following:

Expectation[Exp[x], 
 x \[Distributed] 
  MarginalDistribution[
   TruncatedDistribution[{{0, ∞}, {0, ∞}, {0, ∞}}, 
    MultinormalDistribution[{μa, μb, μc},
      {{s11, s12, s13}, {s12, s22, s23}, {s13, s23, s33}}]], 1]]

Answer 1

Since you're only interested in the $x$ variable, form the Marginal Distribution by (effectively) integrating over $y$ and $z$. Then either do an integration over positive $x$:

Integrate[
 x PDF[MarginalDistribution[
    MultinormalDistribution[{μa, μb, μc}, 
      {{s11, s12, s13}, {s12, s22, s23}, {s13, s23, s33}}], 1], x], 
{x, 0, ∞}]

  (*
    ConditionalExpression[(
     E^(-(μa^2/(2 s11))) s11 + 
      Sqrt[π/2] Sqrt[
       s11] μa (1 + Erf[μa/(Sqrt[2] Sqrt[s11])]))/(
     Sqrt[2 π] Sqrt[s11]), (Re[s11] >= 0 && Re[μa/s11] < 0) || 
      Re[s11] > 0]
  *)

or a one-dimensional Expectation over the new distribution.

Basic sanity check:

Expectation[x, 
 x \[Distributed] 
  MarginalDistribution[
   MultinormalDistribution[{μa, μb, μc}, {{s11, s12, 
      s13}, {s12, s22, s23}, {s13, s23, s33}}], 1]]

  (*
    μa
  *)

So here is nearly the final answer (which does not constrain $y$ and $z$ to be positive):

Expectation[Exp[x] \[Conditioned] x > 0, 
 x \[Distributed] 
  MarginalDistribution[
   MultinormalDistribution[{μa, μb, μc}, 
    {{s11, s12, s13}, {s12, s22, s23}, {s13, s23, s33}}], 1]]

$\frac{e^{\text{$\mu $a}+\frac{\text{s11}}{2}} \left(\text{erf}\left(\frac{\text{$\mu $a}+\text{s11}}{\sqrt{2} \sqrt{\text{s11}}}\right)+1\right)}{\text{erfc}\left(-\frac{\text{$\mu $a}}{\sqrt{2} \sqrt{\text{s11}}}\right)}$

I think to constrain $y$ and $z$ to be positive you'll have to use TruncatedDistribution. (I may have time to solve this tomorrow...)